To solve the quadratic equation \( x^2 + 8x + 7 = 0 \), we will utilize the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are:
- \( a = 1 \)
- \( b = 8 \)
- \( c = 7 \)
Step 1: Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:
[tex]\[
\Delta = 8^2 - 4 \cdot 1 \cdot 7 = 64 - 28 = 36
\][/tex]
Step 2: Compute the two solutions using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since \( \Delta = 36 \), we find:
[tex]\[
x = \frac{-8 \pm \sqrt{36}}{2 \cdot 1} = \frac{-8 \pm 6}{2}
\][/tex]
Step 3: Calculate the two possible values for \( x \):
[tex]\[
x_1 = \frac{-8 + 6}{2} = \frac{-2}{2} = -1
\][/tex]
[tex]\[
x_2 = \frac{-8 - 6}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the quadratic equation \( x^2 + 8x + 7 = 0 \) are:
[tex]\[
x = -1 \quad \text{and} \quad x = -7
\][/tex]
Thus, the correct answer is:
[tex]\[
x = -7 \quad \text{and} \quad x = -1
\][/tex]
